Asymptotic normalization coefficients from proton transfer reactions and astrophysical S factors for the CNO 13C(p, γ)14N radiative capture process

Nuclear Physics A 725 (2003) 279–294 www.elsevier.com/locate/npe

Asymptotic normalization coefficients from proton transfer reactions and astrophysical S factors for the CNO 13C(p, γ )14N radiative capture process A.M. Mukhamedzhanov a , A. Azhari a , V. Burjan b,∗ , C.A. Gagliardi a , V. Kroha b , A. Sattarov a , X. Tang a , L. Trache a , R.E. Tribble a a Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA b Nuclear Physics Institute, Czech Academy of Sciences, Rez near Prague, Czech Republic

Received 19 November 2002; received in revised form 20 June 2003; accepted 25 June 2003

Abstract The 13 C(p, γ )14 N radiative capture reaction is analyzed within the R-matrix approach with the aim to determine the astrophysical S factor. The low-energy astrophysical S factor has important contributions from both resonant and nonresonant captures. The contribution of the nonresonant component of the transition to a particular 14 N bound state is expressed in terms of the asymptotic normalization coefficient (ANC). The experimental ANCs induced from the 13 C(14 N, 13 C)14 N and 13 C(3 He, d)14 N reactions are used in the analysis. The fits of the calculated S factors to the experimental data are sensitive to the ANC values and are used to test the extracted ANCs. It is found that the ANCs determined from the transfer reactions provide the best fit for transitions to all the states in 14 N, except for the third excited state. The obtained astrophysical factor, S(0) = 7.6 ± 1.1 keV b, is in excellent agreement with published values.  2003 Elsevier B.V. All rights reserved. PACS: 25.40.Lw; 24.30.Gd; 26.20.+f; 27.20.+n Keywords: Radiative capture reaction; Asymptotic normalization coefficient; R-matrix approach

* Corresponding author.

E-mail addresses: [email protected] (A.M. Mukhamedzhanov), [email protected] (V. Burjan). 0375-9474/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9474(03)01618-X

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1. Introduction The 13 C(p, γ )14 N reaction belongs to the important reactions in the CNO cycle. It precedes the slowest reaction in the CNO cycle, the 14 N(p, γ )15 O radiative capture reaction, which defines the rate of energy production in the cycle [1]. In addition, the 13 C(p, γ )14 N radiative capture rate is important for nucleosynthesis via the s process because it can deplete the seed nuclei required for the neutron generator reaction 13 C(α, n)16 O in asymptotic giant branch stars with solar metalicity [2,3]. Before 1994, the extrapolation of the measured astrophysical S factors down to zero energy led to a range of S(0) = 5–12 keV b and a recommended value of S(0) = 5.5 keV b [4]. Only capture to the ground state of 14 N was measured; the S factor was renormalized to take into account contributions from possible captures to excited states of 14 N. The most accurate and thorough measurements of the S factor, including captures to the first six excited states of 14 N, were carried out by King et al. in the energy range E = 100–900 keV [5]. Their extrapolation of the total S factor down to stellar energies gave S(0) = 7.64 keV b and S(25 keV) = 7.7 ± 1.0 keV b. This was higher than the recommended value because the contribution from excited states was found to be larger than previously expected. There are two low-lying resonances in 14 N at center-of-mass energies ER1 = 417.9 keV π (J = 2− , T = 0) and ER2 = 511.38 keV (J π = 1− , T = 1), as shown in Fig. 1 [6] (ER2 = 518.14 MeV in [5]). The first resonance is very narrow and does not affect the astrophysically important region. The second resonance dominates the capture cross section in the energy region around and below ER2 down to zero energy. There is another very wide resonance in 14 N at E π − R3 = 1225.7 ± 6.5 keV (J = 0 , T = 1) [6]. Its updated energy is ER3 = 1251.0 ± 7.0 keV [7]. Also its effect on the low energy S factor is small if we took this resonance into account when fitting the experimental data. In the 13 C(p, γ )14 N reaction the first seven bound states of 14 N are populated, proceeding through resonant and nonresonant captures. An accurate determination of the nonresonant contribution is very important for a determination of the total capture rate at stellar energies. In the conventional analysis applied in Ref. [5], the nonresonant capture amplitudes were parametrized in terms of spectroscopic factors which were treated as free parameters. In addition, Ref. [5] used the traditional potential model. However, when it is applied to captures to tightly bound states such as those in 14 N, the capture amplitudes are sensitive to the assumed initial state optical potentials, and antisymmetrization effects, which are typically neglected, are important due to the contribution from small radii. The dependence of the interference between the nonresonant and resonant amplitudes on channel spin was also neglected in the analysis done in Ref. [5]. Here we reanalyze the experimental data of [5] within the framework of the R-matrix approach. We present a more detailed description of the whole procedure which was published earlier [9]. In this approach the resonant and nonresonant contributions are treated in a consistent way, thereby avoiding double counting. The R-matrix nonresonant amplitudes are parametrized in terms of the asymptotic normalization coefficients (ANCs) for 13 C + p ↔ 14 N. We show that this reduces the uncertainty in the nonresonant capture amplitudes to their dependence on the channel radius. It also provides a means to test the experimental values of the ANCs, which were derived in previous proton transfer

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Fig. 1. The 14 N energy level scheme [6]. Only bound states and the first five resonances are shown.

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reaction studies [11,12]. Our analytical form of the astrophysical S factor reproduces the experimental data in the energy interval 105–876 keV (in the c.m.) for the transitions to each bound state. We extrapolate this expression down to zero energy to determine the S(0) and S(25) factors. Ref. [13] describes an alternative fit to the 13 C(p, γ )14 N(gnd) data of King et al. Unlike the results presented here, the fit in Ref. [13] neglects the interference effects between the resonant and nonresonant contributions, so it systematically underestimates the experimental S factor below the second resonance and overestimates it above the resonance.

2. Formalism Let us consider the radiative capture reaction B + b → A + γ . The R-matrix radiative capture cross section to a state of nucleus A with a given spin Jf is given by [14]
σJf = σJi Jf , (1) Ji

σJi Jf =


  π 2Ji + 1 U I l J J 2 . i f i 2 k (2Jb + 1)(2JB + 1)

(2)

I li

Here Ji is the total angular momentum of the colliding nuclei B and b in the initial state, Jb and JB are the spins of nuclei b and B, and I , k, and li are their channel spin, relative momentum and orbital angular momentum in the initial state. UI li Jf Ji is the transition amplitude from the initial continuum state (Ji , I, li ) to the final bound state (Jf , I ). Interference effects only occur in Eq. (2) if the resonant and nonresonant amplitudes have the same channel spin I and orbital angular momentum li . The astrophysical S factor is related to the cross section σ as S(E) = Ee2πηi σ,

(3)

where E is the relative energy, ηi = Zb ZB µBb /k is the Coulomb parameter in the initial state, Zj is the charge of the particle j , and µBb is the reduced mass of particles B and b. UI li Jf Ji is given by the sum of the resonant and nonresonant transition amplitudes. In the one-level, one-channel approximation, the resonant amplitude for the capture into the resonance with energy ERn and spin Ji , and subsequent decay into the bound state with the spin Jf ,  Ji 1/2 J ΓbI li (E)Γγ Ji f (E) i(ωli −φli ) R UI li Jf Ji = −ie . (4) ΓJ E − ERn + i 2 i Here we assume that the boundary parameter is equal to the shift function at resonance Ji energy and φli is the solid sphere scattering phase shift in the li th partial wave, ΓbI li (E) is the observable partial width of the resonance in the channel B + b with the given set of quantum numbers, ΓγJJi f (E) is the observable radiative width for the decay of the given

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resonance into the bound state with the spin Jf , and ΓJi is the observable total width of the resonance level. The phase factor ωli is given by   li
−1 ηi ωli = tan . (5) n n=1

For all the resonances here, ΓJi ≈ is given by J

ΓbIi li (E) = and

I

Ji ΓbI li . The energy dependence of the partial widths

Pli (E) Ji Γ (ERn ), Pli (ERn ) bI li 

J

Γγ Ji f (E) =



E + εf ERn + εf

2L+1

J

Γγ Ji f (ERn ).

(6)

(7)

Ji Ji ΓbI li (ERn ) and Γγ Jf (ERn ) are the experimental partial and radiative widths, εf is the proton binding energy of the bound state in nucleus A, and L is the multipolarity of the gamma quanta emitted during the transition. The penetrability Pli (E) is given by

Pli (E) =

ka Fl2i (k, a) + G2li (k, a)

,

(8)

where Fli and Gli are the regular and singular (at the origin) solutions of the radial Schrödinger equation, and a is the channel radius. The nonresonant amplitude is given by 3/2 li +L−lf +1 i(ωli −φli ) 1 µBb L+1/2 i e UINR li Jf Ji = −(2) k   (L + 1)(2L + 1) Zb e L ZB e × + (−1) L L mL m b B 1 (kγ a)L+1/2CJf I lf Fli (k, a)Gli (k, a)Wlf (2κa) × (2L + 1)!! × Pli (li 0 L0|lf 0)U (Llf Ji I ; li Jf )JL (li lf ),

(9)

where JL (li lf ) =

1 a2

∞ dr r a

 Wlf (2κr) Fli (k, r) Gli (k, r) − . Wlf (2κa) Fli (k, a) Gli (k, a)

(10)

Here, Wl (2κr) is the Whittaker hypergeometric function, κ = 2µBb εf and lf are the wave number and relative orbital angular momentum of the bound state, and kγ = E + εf is the momentum of the emitted photon. We use the system of units such that h¯ = c = 1. The nonresonant amplitude is normalized in terms of the ANC, CJf I lf , which defines the amplitude of the tail of the bound state wave function of nucleus A projected onto the two-body channel B + b with the quantum numbers Jf , I, lf . Such a normalization is physically transparent: in the R-matrix method the internal part of the nonresonant

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amplitude is included in the resonant part while the external nonresonant part contains the tail of the overlap function of the bound wave functions of A, B and b, whose normalization is given by the corresponding ANC. In the conventional R-matrix approach the nonresonant amplitude is normalized in terms of the reduced width amplitude, which is not directly observable and depends on the channel radius. However, it is more convenient to express the normalization of the nonresonant amplitude in terms of the ANC that can be measured independently [15]. Then only the radial matrix element depends on the channel radius. In a strict R-matrix approach the partial width is given by 1/2  1/2  Ji = 2Pli (E) γbI li Ji . (11) ΓbI li (E) Here the reduced-width amplitude γbI li Ji is given by the sum of the internal and external (or channel) reduced-width amplitudes: γbI li Ji = γbI li Ji (int) + γbI li Ji (ch).

(12)

While the internal reduced-width amplitude is real, the channel reduced-width amplitude is complex [16]. In addition, the denominator for the resonant amplitude is complex. Thus, the interference between the resonant and nonresonant terms is neither constructive nor destructive, but rather quite complicated. However, in the 13 C(p, γ )14 N reaction the interference between the resonant and nonresonant amplitudes occurs only for tightly bound states where the channel reduced-width amplitude is small compared to the internal reduced-width amplitude for a channel radius that is somewhat larger than the nuclear radius. Since the best fit has been derived for a channel radius of a = 5 fm, we have neglected the channel reduced-width amplitude when calculating the interference terms. Also we neglected the contribution of the first resonance because it is so narrow that it does not affect the low energy S factor.

3. Analysis of data The nonresonant transitions to the ground and all first six excited bound states are dominated by E1 transitions [5]. The second resonance, (1− , 1), can decay into the ground and first five excited states in 14 N, thus contributing to the corresponding resonant captures [5, 6]. The fifth resonance, 0− , 1, can decay into the ground and four excited states in 14 N, thus contributing to the corresponding resonant captures. The parameters of the second and fifth resonances were allowed to vary in the intervals reported in [5,6] and [7], correspondingly. A crucial fitting parameter in the R-matrix method is the channel radius a. We find that a = 5.0 fm provides the best fit for the transitions to the ground and first five excited states. For the transitions to the sixth excited state, we find a better fit with a = 4.0 fm. In a strict R-matrix approach, the interference of the resonant and nonresonant amplitudes contains

Ji Ji ΓbI li (E) [14]. Knowledge of the observed resonance width ΓbI li (E) is not sufficient

Ji to calculate ΓbI li (E). However, interference between the resonant and nonresonant amplitudes occurs only for the transitions to the ground, first and second excited states. The high binding energies and large channel

radius for these transitions allows us to neglect the

contribution of the channel part of

Ji ΓbI li (E) in the interference term. Then a knowledge

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Table 1 Resonance parameters Jπ,T Ref. [6] 1− , 1 0− , 1

ER (keV) Ref. [6] 511.64±0.90 1225.70±6.50

Ref. [5]

Ref. [7]

Present

518.14±0.95 1225.70±6.50

1251.0±7.0

517.0 1244.0a

37.04 408.6±7.4

416.0a

Γp (keV) 1− , 1 0− , 1

29.72±0.93 408.6 ± 20.4

37.14±0.98 408.57

a The uncertainty of the resonance energy in the “present” column is the same as in the previous column.

Table 2 Radiative widths. The first column gives the resonance energy which provides the best fit. Ef is the excitation energy of the bound state in 14 N. The uncertainties of the radiative widths in the “present” column are the same as in the previous column Resonance

Transition resonance to bound state

Jiπ , ER (MeV)

Jfπ , Ef (MeV)

1− , 0.52

1+ , 0+ , 1+ , 0− , 2− , 1− ,

0.00 2.31 3.95 4.92 5.11 5.69

0− , 1.251a

1+ , 1+ , 2− , 1− ,

0.00 3.95 5.11 5.69

J

Radiative width Γγ Ji (eV) f

Ref. [6]

Ref. [5]

Present

9.9 ± 2.5 0.17±0.05 1.56±0.40 0.23±0.06 0.03±0.02 0.43±0.12

9.10 ± 1.30 0.22 ± 0.04 1.53 ± 0.21 0.262±0.043 0.075±0.025 0.61 ± 0.14

9.10 0.22 1.53 0.260 0.085 0.63

46 ± 12

46 ± 12 0.46 ± 0.14a 0.37 ± 0.14a 0.23 ± 0.20a

56 0.56 0.23 0.23

a Ref. [8].

 J  of ΓbIili (E) is sufficient to calculate the interference term. For the transitions to excited states beyond the second excited state, the S factor is given by the incoherent sum of the resonant and nonresonant captures, and hence is expressed in terms of the observable resonance widths. Other fitting parameters are the resonance energies, widths, radiative widths and ANCs. The resonance energies, widths and radiative widths for transitions to each bound state were varied within the intervals determined in Ref. [5] and are given in Tables 1 and 2. In principle the ANCs can be inferred directly from the fit to the experimental S factors for each transition. However, the uncertainties of the ANCs determined from such fits are quite large due to the experimental uncertainties and resonance contributions. In this paper we use another approach. As noted above, the ANC for the synthesis B + b → An leading to the nth bound state of nucleus A defines the absolute normalization of the nonresonant capture S factor to this state. The ANCs for the captures to the ground and first six excited

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Table 3 The ANCs for 13 C + p → 14 N. The Jfπ , Ef are given in the first column for states in 14 N; the corresponding proton orbital and total angular momenta, lf and jf are given in the second column. The ANCs determined from the 13 C(14 N, 13 C)14 N and 13 C(3 He, d)14 N reactions [11,12] are given in the third column. The superscripts ‘a’ and ‘b’ denote the ANCs determined from the 14 N and 3 He induced reactions, respectively, while the superscript ‘a,b’ indicates those that come from the average of the two experiments. The ANCs providing the best fit are presented in the last column. All the ANCs here are given in jj -coupling, while in the equation for the S factor the channel spin representation has been used. With the exception of the 0− state, the uncertainties of the ANCs from the fits are significantly worse than those from the transfer reactions and, hence, are not shown in the last column State 14 N Jfπ , Ef (MeV) 1+ , 0.00 0+ , 1+ , 0− , 2− , 1− , 3− ,

2.31 3.95 4.92 5.11 5.69 5.83

Proton orbitals lf j

ANCs [11,12] C 2 (fm−1 )

Present C 2 (fm−1 )

p1/2 p3/2 p1/2 p1/2 s1/2 d5/2 s1/2 d5/2

18.2 ± 0.9a,b 0.91 ± 0.14a,b 8.9 ± 0.9a 2.88 ± 0.17a,b 12.66 ± 0.89b 0.40 ± 0.03a 10.33 ± 0.72b 0.19 ± 0.02a

18.2 0.91 8.9 2.71 33.00 ± 3.81 0.43 10.95 0.21

f

a Ref. [11]. b Ref. [12].

states in 14 N have been measured in the 13 C(14 N, 13 C)14 N [11] and 13 C(3 He, d)14N [12] proton transfer reactions. The ANCs determined from these transfer reactions are given in the third column of Table 3. We use these ANCs to calculate the S factors for each transition. It allows us to check the ANCs determined independently in the analysis of the astrophysical factors. The ANCs for each state were varied within the uncertainty intervals found from the transfer reactions. For all the states, except the third excited state with Ef = 4.92 MeV, the best fit was provided by the ANCs determined from the nonresonant proton transfer reactions (Table 3). The results for the ANCs providing the best fit are given in the last column of Table 3. Since the uncertainty intervals for the ANCs found from the transfer reactions are significantly smaller than the uncertainties for the ANCs determined from the fit to the data of Ref. [5], the uncertainties shown in column 3 of Table 3 are those found from the transfer reactions except for the third excited state. The values shown in the fourth column of Table 3 correspond to the ANCs which give the best fit to the data of Ref. [5] when the ANCs are allowed to vary within the uncertainty interval determined from the transfer reactions [11,12] with, again, the exception of the third excited state. The best fit for the transition to the third excited state with Ef = 4.92 MeV requires an ANC exceeding that found in Ref. [12] by approximately a factor of 2.6. The ANC for this state and its uncertainty determined from the fit are given in Table 3. To estimate the uncertainties of the fits, resonance parameters were varied within the intervals determined in [5], Tables 1 and 2, and the ANCs were varied within the narrow intervals determined from the transfer reactions, Table 3. The channel radius has been changed by 20% from the central value r0 = 5 fm which provides the best fit. We also

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took into account the 11.3% systematic normalization uncertainty in the experimental data reported in [5]. 3.1. S factor for the transition to the ground state The ground state of 14 N has J π = 1+ , T = 0, I = 0, 1 and lf = 1, and a proton binding energy of ε0 = 7.55 MeV. Nonresonant capture to the ground state proceeds by an E1 transition with li = 0 → lf = 1. Both resonances, (1− , 1), and (0− , 1) also decay to the ground state via E1 transitions. Since the resonant and nonresonant capture amplitudes to both resonances have the same channel spins, they interfere. The parameters used in the R-matrix fit are given in Tables 1, 2 and 3. The results of the fit are displayed in Fig. 2(a). The fit gives S(0) = 5.16 ± 0.72 keV b. Although the nonresonant S factor gives a small contribution at zero energy, S (NR) (0) = 0.32 keV b, its contribution is important due to interference with the resonant amplitude. The calculated resonant S factor at zero energy is S (R) (0) = 3.74 keV b. Thus, the nonresonant contribution at zero energy is 8.6% of the resonant one, but the interference contribution is S (INT) (0) = 1.10 keV b or 21% of the total. The uncertainty of the fit takes into account the uncertainties of the resonance parameters determined in [5], ANCs, the channel radius and experimental systematic and statistical uncertainties. There are only two parameters to be varied in the nonresonant part: the ANCs and the channel radius. The variation of the ANCs by 6% and the channel radius by 20% changes S(0) less than 1%; however the shape of the high energy wing changes more significantly, thus restricting the variation of these parameters. We estimate the uncertainty in S(0) for the ground state transition due to the nonresonant contribution to be less than 1%. An additional uncertainty of 5% is assigned to the S(0) due to the uncertainties of the parameters of the fifth resonance 0− , 1. The inclusion of an s-wave background pole, ER = 7 MeV, Γp = 3 MeV and Γγ = 5 eV, which interferes with the direct, second and fifth resonance terms, slightly improves the fit. An additional uncertainty of 9% has been included in the extrapolation of the total S factor to zero energy due to the background pole. 3.2. S factors for transitions to excited states Capture through the second resonance, (1− , 1), to six excited states in 14 N contributes to the total S factor. Capture through the fifth resonance (0− , 1) contributes to captures to four excited states, second, fourth, fifth and sixth [8], but capture to the sixth excited state was too weak to be detected in [5] and, hence, is not analysed here. Relevant quantum numbers for each state are given in Tables 2 and 3. With the exception of the first excited state, which has isospin T = 1, all transitions are to T = 0 final states. The nonresonant capture to the first excited state is an E1 transition with li = 0 → lf = 1. The ANCs extracted for this state from the 13 C(14 N, 13 C)14 N [11] and 13 C(3 He, d)14N [12] proton transfer reactions differ by nearly a factor of 2. Similar problems have been encountered in extracting the ANCs from the 9 Be(10 B, 9 Be)10 B and 9 Be(3 He, d)10 B reactions populating the second excited state (Ex = 1.74 MeV, T = 1) in 10 B [17]. We attempted fitting the experimental data with both values of the ANC. The results are presented in Fig. 2(b). The ANC, C 2 = 8.9 ± 0.9 fm−1 , determined from the heavy ion reaction [11] provides a

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Fig. 2. 13 C(p, γ )14 N astrophysical S factors for the captures to the ground and first three excited states. In all figures the squares are the data points [5] and dotted lines the nonresonant capture S factors only. (a) The ground state (1+ , 0.00 MeV); The solid line is the best fit which includes the resonant terms (the second, fifth and background resonances) and the nonresonant capture terms corresponding to two channel spins and their interference. (b) The first excited state (0+ , 2.31 MeV); the solid line and dashed-dotted lines are the fits calculated using the ANCs C 2 = 8.9 fm−1 [11,12] and C 2 = 16.0 fm−1 [12], respectively; the two dotted lines are the nonresonant capture S factors calculated with the ANC C 2 = 8.90 fm−1 (wide spacing) [11,12] and C 2 = 16.0 fm−1 [12] (narrow spacing). (c) The second excited state (1+ , 3.948 MeV); notations the same as for (a). (d) The third excited state (0− , 4.92 MeV). The solid line and dashed-dotted lines are the fits calculated using the ANCs C 2 = 33.0 fm−1 and C 2 = 12.66 fm−1 Ref. [12], respectively. Both fits include the contributions from the second resonance and nonresonant terms. The two dotted lines are the nonresonant capture S factors calculated with the ANC C 2 = 33.0 fm−1 (wide spacing) and C 2 = 12.66 fm−1 (narrow spacing) Ref. [12].

better fit than C 2 = 16.0 ± 1.1 fm−1 determined from the (3 He, d) reaction [12], especially at energies above the resonance. The χ 2 for the fit with the lower ANC is also only half that for the fit with the larger ANC. We conclude that the lower value is more appropriate and obtain for the transition to the first excited state S(0) = 0.32 ± 0.08 keV b, with the nonresonant term contributing about 33% to the total S factor. The nonresonant capture to the second excited state proceeds through an E1 transition with two channel spins, I = 0, 1. Both resonances, (1− , 1), and (0− , 1) also decay to the second excited state via E1 transitions with channel spins I = 1 and I = 2,

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correspondingly. The total S factor is given by the sum of the S factors with channel spins I = 0 and I = 1, where the I = 1 contribution is a coherent sum of the second and background resonant amplitudes and nonresonant amplitude, while the I = 0 contribution is a coherent sum of the fifth resonant and nonresonant amplitude. The result of the S factor fit for the transition to the second excited state is displayed in Fig. 2(c). The fit for the transition to the second excited state gives a total S(0) = 0.90 ± 0.13 keV b, with S (NR) (0) = 0.27 keV b, S R (0) = 0.36 keV b and S (INT) (0) = 0.27 keV b. The inclusion of the s-wave background pole described above improves the overall fit. The ground state and first two excited states are positive parity and both resonant and nonresonant capture to them are dominated by E1 transitions. The remaining four excited states are all negative parity. The nonresonant capture transitions to these states proceed through E1 transitions, but the resonant captures through the second, (1− , 1), and fifth, (0− , 1), resonances proceed through either M1 or E2 transitions. Hence there is no interference between the direct and resonant parts. Generally the fits are very similar assuming either M1 or E2 transitions, making it difficult to distinguish between the two. In what follows we present only fits with M1 transitions. The results of the fit to the third excited state are shown in Fig. 2(d). In contrast to the previous transitions, the nonresonant capture S factor increases with energy [10]. Only the second resonance contributes to the S factor for capture to the third excited state. The resonant S factor drops sharply when moving away from the resonance peak. Hence nonresonant capture dominates over resonance capture on both the low-energy and high-energy wings. This state was weakly populated in the 13 C(14 N, 13 C)14 N reaction due to the angular momentum mismatch and could not be resolved from the fourth excited state [11]. However, it was resolved in the 13 C(3 He, d)14N reaction and an ANC, C 2 = 12.66 ± 0.89 fm−1 [12], was extracted assuming a 2s1/2 configuration for the transferred proton in 14 N. The behavior of the S factor makes it possible to determine the ANC for this state from the fit. The best fit from the data and the fit using the ANC from Ref. [12] are both shown in Fig. 2(d). The best fit is achieved with C 2 = 33.0 fm−1 . The uncertainty of the ANC determined from the data is ∼12%. With this value, the total S factor for the transition to the third excited state at zero energy is S(0) = 0.33 ± 0.07 keV b, with S (NR) (0) = 0.28 keV b and S (R) (0) = 0.06 keV b. Both the second and fifth resonances can contribute to the S factor for the fourth excited state. The fits are very similar assuming either M1 or E2 transitions, making it difficult to distinguish between the two. We obtain an excellent fit to the data, as shown in Fig. 3(a), including nonresonant capture, with a magnitude given by the ANC from [11], and taking into account the background pole which interferes with the second resonance destructively at energies E < 0.517 MeV. The resonant and nonresonant terms do not interfere. The calculated total S factor for this transition is S(0) = 0.046 ± 0.009 keV b, where S (NR) (0) = 0.027 keV b and S (R) (0) = 0.019 keV b. As with the third excited state, the ANC for the fifth excited state was obtained only from the data measured in the 13 C(3 He, d)14 N reaction. The ANC value, C 2 = 10.33 ±0.72 fm−1 , was found assuming a 2s1/2 configuration for the transferred proton in 14 N [12]. Varying the ANC within the uncertainty range obtained from the transfer reaction we calculated the direct and resonance contributions to the S factor for capture to the fifth excited state. We find that C 2 = 10.95 fm−1 provides the best fit. Thus, in contrast to the

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Fig. 3. The 13 C(p, γ )14 N astrophysical S factors for the captures to the fourth, fifth and sixth excited states, and the total low-energy S factor. In all figures the squares are the data points [5] and dotted lines the nonresonant capture S factors only. (a) The fourth excited state (2− , 5.11 MeV); The solid line is the best fit which represents the noncoherent sum of the resonant (the second, fifth and background resonances) and nonresonant S factors. (b) The fifth excited state (1− , 5.691 MeV). The solid line is the best fit which represents the noncoherent sum of the second and fifth resonant and the nonresonant S factors. (c) The sixth excited state (3− , 5.834). The solid line is the nonresonant capture S factor. (d) The total low-energy astrophysical S factor for the captures to the ground and first six excited states in 14 N. The solid line is the fit.

third excited state, the ANC for the fifth excited state obtained from the transfer reaction gives a good accounting of the direct capture for transition to this state. The results of the best fit are shown in Fig. 3(b). Both the second and the fifth resonances contribute to the S factor for capture to the fifth excited state. As with the fourth excited state, the fits are very similar assuming either M1 or E2 transitions, making it difficult to distinguish between the two. The calculated total S factor for the transition to the fifth excited state is S(0) = 0.77 ± 0.09 keV b, with S (NR) (0) = 0.66 keV b and S (R) (0) = 0.11 keV b. The sixth excited state is the highest energy level of 14 N populated in the 13 C(p, γ )14 N radiative capture process. Very little is known about this transition. The ANC for it provides

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a measure of the nonresonant capture into this state. Fig. 3(c) shows the data available for this transition compared to the S factor based only on nonresonant capture. A channel radius of 4.0 fm provides a better fit to the data than the 5.0 fm that was used for the other transitions. However, a comparable fit can be obtained with a channel radius of 5.0 fm if we include an internal contribution to the transition amplitude. The calculated nonresonant capture S factor for this transition is S(0) = 0.031 ± 0.007 keV b. Hence it contributes less than 0.5% to the total S(0) factor. 3.3. Total S factor The total S factor is the sum of the individual S factors for the transitions to the ground and first six excited states. Our result for the low energy part of the total S factor is shown in Fig. 3(d). The value obtained for the total S factor at zero energy is S(0) = 7.60 ± 1.10 keV b. This is in excellent agreement with the result found in the analysis by King et al. [5]. However, it is important to note that our result is obtained by extrapolating the appropriate analytic form of the S factor for each final state to zero energy. In contrast, King et al. used an empirical parametrization of the energy dependence of the S factor to obtain their extrapolation to zero energy. We find that the individual contributions to the total S factor are S (NR) (0) = 1.54 keV b, S (R) = 4.36 keV b and S (INT) (0) = 1.70 keV b. At E = 25 keV the total calculated S factor is S(25) = 8.0 ± 1.2 keV b. The uncertainty for S(0) and S(25) in our analysis come from two sources. In our analysis we use the resonance parameters as determined in [5,7,8]. The ANCs which define the absolute normalization of the direct capture terms are inferred from the transfer reaction measurements and have significantly smaller uncertainties, 6%, than the corresponding spectroscopic factors determined in [5]. We take into account both, the 11.3% systematic and statistical uncertainties of the experimental data for each transition. We find that it leads to an 8.5% uncertainty in the total S factor at energies of 0 and 25 keV. There is an additional uncertainty in the low-energy S factor when we take into account the background in the form of the distant broad resonance which may interfere with the corresponding direct and resonance terms. The inclusion of such a resonance slightly improves the overall fit. The effect of the background brings an additional 9% uncertainty to the total S factor. Hence the total uncertainty of our analysis is 14%, which is close to the total uncertainty claimed in [5]. Our total low-energy S factors are given in Table 4. The polynomial fit of the calculated S factor at 0 < E < 300 keV converges for the polynomial power n = 8, in contrast to [5] where n = 2 was found to be enough. The Taylor series of the calculated total S factor in powers of E, applied for the energy span 0 < E < 300 keV, reads as 
1 dn S   S(E) = En. (13) n! dE n E=0 n=0

For nmax = 8, S(0) = 7.58 keV b and dS/dE|E=0 = 9.5 × 10−3 b and d2 S/dE 2 |E=0 = 4.39 × 10−4 b/keV.

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Table 4 The low-energy astrophysical factor for 13 C + p → 14 N determined in the present work from fitting the experimental astrophysical factor [5]. The first column gives the relative p–13 C kinetic energy in keV, and the second column is the astrophysical factor with its uncertainty E (keV)

Astrophysical S factor (keV b) 7.6 ± 1.1 7.6 ± 1.1 7.7 ± 1.1 8.0 ± 1.2 9.8 ± 1.4 10.9 ± 1.6 11.3 ± 1.7 12.2 ± 1.8 13.5 ± 2.0 14.1 ± 2.0 15.3 ± 2.2

0.1 1 10 25 104.4 140 152.1 173.9 199.6 208.9 227.3

E (keV)

Astrophysical S factor (keV b)

244.1 246.6 249.2 276.4 277.6 279.8 294.4 323 332.4 369.7 385.7

16.6 ± 2.4 16.8 ± 2.4 17.1 ± 2.5 19.9 ± 2.9 20.0 ± 2.9 20.3 ± 2.9 22.3 ± 3.2 27.4 ± 3.8 29.6 ± 4.2 42.6 ± 6.0 51.6 ± 7.2

The reaction rates NA σ v are given by a standard expression: NA σ v = NA

(8/π)1/2

∞

1/2

µBb (kB T )3/2

dE σ Ee−E/(kB T ) ,

(14)

0

where NA is the Avogadro number, v relative velocity of the colliding nuclei, kB the Boltzmann constant, T the temperature, σ the radiative capture cross section. The reaction rates for 13 C(p, γ )14N have been calculated using NACRE code [18] for temperatures T9 < 0.1, which define the hydrostatic hydrogen burning regime, are presented in Table 5. Note that for temperatures T9 < 0.1 only the first three terms of the Taylor expansion are needed to calculate the reaction rates. We also present the ratio of our adopted reaction rates and the reaction rates adopted in the compilation [19]. This ratio is about 1.12, i.e., our adopted reaction rates are in good agreement with those calculated in [5] and slightly higher than those given in compilation [19]. But we obtain higher reaction rates of 13 C consuming than in [4].

4. Summary We have reanalyzed the experimental data for the 13 C (p, γ )14N radiative capture at energies 100 < Ec.m. < 900 keV [5] using the R-matrix approach and experimentally determined ANCs, and then extrapolated down to zero energy. The R-matrix method correctly accounts for the interference between the nonresonant and resonant amplitudes and the ANCs define the nonresonant capture rate in a model independent way. The results are sensitive to the ANCs, and thus they provide a consistency check on the ANC values derived from two different proton transfer reactions. The ANCs determined from the 13 C(3 He, d)14N reaction provide good fits to the 13 C(p, γ )14 N S factors populating the ground state and the second and fifth excited states, but are not consistent with the

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Table 5 Low-temperature reaction rates in cm3 mole−1 s−1 for 13 C + p → 14 N calculated using our astrophysical S factors given in Table 4. First column: temperature in 109 K; the second, third and fourth columns are our low, adopted and high reaction rates; the last column gives the ratios of our reaction rates to the adopted reaction rates in compilation [19] T9

Low

0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.018 0.02 0.025 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1.89E-22 3.93E-21 5.12E-20 4.67E-19 3.21E-18 1.78E-17 8.18E-17 3.25E-16 1.14E-15 3.57E-15 2.69E-14 1.54E-13 5.05E-12 7.22E-11 3.49E-09 5.512E-08 4.53E-07 2.45E-06 9.83E-06 3.21E-05 8.89E-05

Adopted reaction rates 2.21E-22 4.60E-21 5.99E-20 5.46E-19 3.76E-18 2.08E-17 9.57E-16 3.80E-16 1.33E-15 4.17E-15 3.15E-14 1.80E-13 5.91E-12 8.45E-11 4.08E-09 6.45E-08 5.30E-07 2.86E-06 1.15E-05 3.756E-05 1.04E-05

High

Ratio

2.53E-22 5.27E-21 6.86E-20 6.25E-19 4.31E-18 2.38E-17 1.10E-16 4.35E-16 1.52E-15 4.77E-15 3.61E-14 2.06E-13 6.77E-12 9.68E-11 4.67E-09 7.39E-08 6.07E-07 3.27E-06 1.32E-05 4.29E-05 1.19E-04

1.11 1.11 1.12 1.12 1.12 1.12 1.12 1.12 1.13 1.13 1.13 1.13 1.13 1.13 1.12 1.12 1.11 1.11 1.10 1.09 1.08

astrophysical S factors for the first and third excited states. The discrepancy for the 14 N first excited state has been noted before [12], and may be due to its J π = 0+ , T = 1 character. It is interesting to note that the 14 N third excited state is the only other spin-0 final state. A possible explanation of the problem with the determination of the ANCs for the states with J = 0 from the (3 He, d) reactions was discussed in [12]. In contrast, we find that the ANCs determined from the 13 C(14 N, 13 C)14 N reaction provide good fits to the 13 C(p, γ )14 N S factors in all six cases for which the ANCs were measured. Our results for the S factors are in excellent agreement with those obtained by King et al. [5] despite the differences in the analyses. The S factor that we obtain, S(0) = 7.6 ± 1.1 keV b, is significantly higher in the hydrostatic hydrogen burning temperature range than the value given previously by Caughlan and Fowler [4]. This higher rate reduces the amount of 13 C available for the 13 C(α, n)16O reaction, thus making it less effective as a neutron source to fuel heavy element synthesis through the s process.

Acknowledgements This work was supported in part by the US Department of Energy under Grant No. DE-FG03-93ER40773, the US National Science Foundation under Grant No. INT-

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9909787 and Grant No. PHY-0140343, ME 385(2000) project NSF and MSMT, CR, grant GACR 202/01/0709, and by the Robert A. Welch Foundation.

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